We now introduce an important operation of graphs,
which is known as the celebrated Kelmans operation;
see, e.g., [1]} or [2]}.
Let \(G\)  be a graph and \(u,v\in V(G)\)  be distinct vertices.
We define a new graph \(G^*\)  by replacing the edge
\(\lbrace v,x\rbrace \)  by a new edge \(\lbrace u,x\rbrace \)  for all \(x\in N(v) \setminus (N(u)\cup \lbrace u\rbrace )\) ,
and all vertices different from \(u, v\)  remain unchanged.
Note that vertices \(u,v\)  are adjacent in \(G^*\)  if and only if
they are adjacent in \(G\) .
An isomorphic graph is obtained if the roles of \(u\)  and \(v\)  are interchanged.
Furthermore, Csikvári [3]} showed that the Kelmans operation
increases the spectral radius of a graph.
A variant of this operation can be seen in [4]}.
